On the chromatic tower

Abstract

We fix a prime p and work in the p - local stable homotopy category. Then, Hopkins' chromatic splitting conjecture essentially predicts the information of a p - completed finite spectrum, obtained using the first ( n + 1) Morava K - theories K (0), K (1), . . . , K ( n ), may be obtained using a single higher Morava K - theory K ( m ). ( m ≥ n +1). However, in spite of its importance, this conjecture is very difficult and subtle. Actually, Devinatz noted the conjecture is false, as soon as we omit the finiteness assumption to include such a nice infinite spectrum as the p - completed BP spectrum. In this paper, we prove a result which reconciles Hopkins' chromatic splitting conjecture and Devinatz' observation about the p - completed BP spectrum. For the p - completion of "nice" spectra, including finite spectra and the BP - spectrum, our result essentially claims that the information obtained using the first ( n +1) Morava K - theories K (0), K (1), . . . , K ( n ), may be obtained using any m - k consecutive higher Morava K - theories K ( k + 1), K ( k + 2), . . . , K ( m - 1), K ( m ) with m - k ≥ n + s 0 + 1. Here, n 0 is the Hopkins-Ravenel (Hovey-Sadofsky) uniform horizontal vanishing line for the E ( n )-based standard Adams-Novikov spectral sequence.